Prime Tuples

Prime tuples are pairs, triples, quadruples ... from neighboring prime numbers. In a broader sense, pairs of the form (p,p+4) or (p,p+6) are also included. The program determines all in an interval [a, b]

Prime twins:
These are prime number pairs of the form (p,p+2), i.e. with the difference 2.

Prime cousins:
These are prime number pairs of the form (p,p+4), i.e. with the difference 4.

Sexy primes:
These are prime number pairs of the form (p,p+6), i.e. with the difference 6.

Prime triplets:
These are triplet primes of the form (p, p+2, p+6) or the form [p, p+4, p+6].
In the program, the two forms are distinguished by the brackets and counted separately.
The triplet (3/5/7) are the only triplets of the form (p, p+2, p+4), since for p> 3 one of the three numbers is always divisible by 3 .

example 1

Prime twins between 1 and 200

(3|5) (5|7) (11|13) (17|19) (29|31) (41|43) (59|61)    
(71|73) (101|103) (107|109) (137|139) (149|151) 
(179|181) (191|193) (197|199) 

15 pairs of prime twins

Example 2

Prime number triplets between 1 and 200

(3|5|7) (5|7|11) [7|11|13] (11|13|17) [13|17|19]   
(17|19|23) [37|41|43] (41|43|47) [67|71|73] 
[97|101|103] (101|103|107) [103|107|109] 
(107|109|113) (191|193|197) [193|197|199] 

15 triplet prime triplets
7 of the form (p|p+2|p+6) and 7 of the form [p|p+4|p+6]

See also:

Prime factorization
Wikipedia: Prime number | Integer factorization